Rhombus tests (3 routes)

Dependencies: SAS congruence, Parallelogram properties (opposite sides equal, diagonals bisect each other), Rhombus: diagonals ⊥-bisect.

Statement

Let $ABCD$ be a quadrilateral (vertices listed in order). The following three conditions are mutually equivalent:

(a) Four equal sides: $|AB| = |BC| = |CD| = |DA|$ (this is the definition of a rhombus); (b) Diagonals $\perp$-bisect each other: the diagonals $AC$, $BD$ meet at $O$, and $AC \perp BD$, $|AO| = |OC|$, $|BO| = |OD|$; (c) A parallelogram with one pair of equal adjacent sides: $ABCD$ is a parallelogram and $|AB| = |BC|$.

Any of the three may serve as a criterion for "$ABCD$ is a rhombus".